In many cases, describing symmetries of quantum systems by proper unitary representations is too restrictive.
Since the measurement statistics are insensitive to global phases, two unitaries, which differ only by a phase, cannot be distinguished by measuring the isolated system.
More concretely, what really matters is the implemented unitary channel \(\mathcal {U}:\rho \mapsto U \rho U^*\)  with \(\rho \in {\mathfrak {S}}(\mathcal {H})\) , where \({\mathfrak {S}}(\mathcal {H})\)  denotes the set of density operators on \(\mathcal {H}\) .
Recall that a density operator is a positive operator with unit trace.
\(\mathcal {U}\)  is an example of a quantum channel [1]}. Quantum channels are completely positive trace preserving (CPTP) maps on the Banach space \(\mathcal {T}*{\mathcal {H}}\)  of trace class operators on \(\mathcal {H}\) .
It is noteworthy that every quantum channel, which has an inverse channel, is already a unitary channel. This follows from Wigner's Theorem [2]} and the fact that conjugation by anti-unitary operators is not completely positive.
